
%!TEX program = xelatex
%!TEX TS-program = xelatex
%!TEX encoding = UTF-8 Unicode

\documentclass[10pt]{article} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% 一些常用的包总结在另一个文件里
\input{wang_preamble.tex}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% 选择Windows操作系统写中文文档，使用 xelatex 或 lualatex 编译器
%\usepackage{xeCJK} % 处理中文、日文和韩文（统称为 CJK 文字）的排版
%\setCJKmainfont{SimSun} % 设置正文字体为宋体
%\setCJKmonofont{SimHei} % 设置加粗字体为黑体
%\setCJKsansfont{SimHei} % 黑体

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%选择Mac操作系统写中文文档，使用 xelatex 或 lualatex 编译器
\usepackage{xeCJK} % 支持中文字体
\setCJKmainfont{Songti SC} % 设置主要中文字体，用于正文中的中文文本。设置主要中文字体为宋体
%\setCJKmonofont{Menlo} % 设置等宽中文字体，用于代码块、等宽文本等。设置等宽中文字体为 Menlo
%\setCJKsansfont{PingFang SC} % 设置无衬线中文字体，用于标题、图表标签等。设置无衬线中文字体为 PingFang SC
%\setCJKromanfont{Songti SC} % 设置罗马中文字体，用于罗马字体中的中文文本。设置罗马中文字体为宋体

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%选择输出文档的两种类型：
\newcommand{\showsolution}{0} %%设置showsolution=0, 编译生成试卷
%\newcommand{\showsolution}{1} %%设置showsolution=1, 编译生成试卷解答

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% 填写课程信息：
\newcommand{\CourseName}{复变函数作业2ABC}
\newcommand{\CourseStudents}{王立庆（2022 级数学与应用数学1班）}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%考完发给学生：

\usepackage{titling}
\setlength{\droptitle}{-2cm}   % 标题上移2cm


\ifnum\showsolution=0
\title{\CourseName }
\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
\renewcommand{\today}{\number\year \,年 \number\month \,月 \number\day \,日}
\date{2024年9月26日}
\fi

\ifnum\showsolution=1
\title{\CourseName \,\, 解答 }
\author{\CourseStudents}
\renewcommand{\today}{\number\year \,年 \number\month \,月 \number\day \,日}
\date{2024年9月26日}
\fi


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\maketitle

\thispagestyle{fancy} % 第一页也显示“第几页，共几页”的信息。
%\thispagestyle{empty} % 第一页不显示页码

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Concept of Analytic Function}

\begin{enumerate}

\item (***)  %%1
Prove that a real function of a complex variable either has the derivative zero, or else the derivative does not exist.

\item (***)  %%2
Prove that an analytic function $f(z)$ is necessarily continuous.

\item (***)  %%3
Let $f(z)=u+iv$ be an analytic function. 
Shows that $|f'(z)|^2$ is the Jacobian of $u$ and $v$ with respect to $x$ and $y$.

\item  %%4
If $u(x,y)$ and $v(x,y)$ have continuous first-order partial derivatives which satisfy the Cauchy-Riemann differential equations, then $f(z) = u(z) + iv(z)$ is analytic with continuous derivative. 

\item (***)  %%5
Find the most general harmonic polynomial of the form $ax^3+bx^2y+cxy^2+dy^3$. 
Determine the conjugate harmonic function and the corresponding analytic function by integration and by the formal method.

\item  %%6
Prove Lucas's theorem. If all zeros of a polynomial $P(z)$ lie in a half plane, then all zeros of the derivative $P'(z)$ lie in the same half plane.

\item (***)  %%7
Use the method of the text (looking for the singular part at each pole) to develop
$$
\frac{z^4}{z^3-1} \,\,\,\,\mathrm{and}\,\,\,\, \frac{1}{z(z+1)^2(z+2)^3}
$$
in partial fractions.

\item  %%8
If $Q$ is a polynomial with distinct roots $\alpha_1,\cdots,\alpha_n$, and if $P$ is a polynomial of degree $<n$, show that 
$$
\frac{P(z)}{Q(z)} = \sum\limits_{k=1}^{n} \frac{P(\alpha_k)}{Q'(\alpha_k)(z-\alpha_k)}. 
$$

\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Elementary Theory of Power Series}

\begin{enumerate}

\item  %%1
Prove the following relations about limes superior and limes inferior, 
\begin{equation*}
\underline{\lim}\, (\alpha_n + \beta_n)
\le \underline{\lim}\, \alpha_n + \overline{\lim}\, \beta_n. 
\end{equation*}

\item (***)  %%2
Show that the following convergence 
$$
\lim\limits_{n\to\infty} \left(1 + \frac{1}{n} \right) x = x
$$
is not uniform for $x$ in $\mathbb{R}$. 

\item  %%3
Prove that the limit function of a uniformly convergent sequence of continuous functions is itself continuous.

\item  %%4
Discuss completely the convergence and uniform convergence of the sequence 
$\{nz^n\}_1^\infty$. 

\item (***)  %%5
Given a power series 
$$ a_0+a_1z+a_2z^2+\cdots+a_nz^n+\cdots. $$
Let $R$ is chosen according to the formula
$$ 1/R = \underset{n\to\infty}{\mathrm{limsup}}\, \sqrt[n]{|a_n|}. $$ 
Assume $0\le \rho < R$. 
Prove that the series converges uniformly for $|z|\le \rho$. 

\item  %%6
Expand $(1-z)^{-m}$, $m$ a positive integer, in powers of $z$.

\item (***)  %%7
If $\lim\limits_{n\to\infty} \frac{|a_n|}{|a_{n+1}|} = R$, prove that $\sum a_nz^n$ has radius of convergence $R$.

\item  %%8
If $\sum\limits_{0}^{\infty}a_n$ converges, then $f(z) = \sum\limits_{0}^{\infty}a_nz^n$ tends to $f(1)$ as $z$ approaches 1 in such a way that $\frac{|1-z|}{1-|z|}$ remains bounded. 

\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Exponential and Trigonometric Functions}

\begin{enumerate}

\item (***)  %%1
Show that the series
\begin{equation*}
e^z = 1 + \frac{z}{1!} + \frac{z^2}{2!} + \cdots + \frac{z^n}{n!} + \cdots 
%\label{eq-}
\end{equation*}
converges in the whole plane. 

\item (***)  %%2
Define the exponential function $e^z$ as the solution of the differential equation
$%\begin{equation*}
f'(z) = f(z)
%\label{eq-}
$%\end{equation*}
with the initial value $f(0)=1$. Show that $e^z$ satisfies the addition theorem
$%\begin{equation*}
e^{a+b} = e^a\cdot e^b. 
%\label{eq-}
$%\end{equation*}

\item (***)  %%3
Show that the addition formulas for the cosine and sine functions 
\begin{equation*}
\begin{aligned}
\cos (a+b) &= \cos a\, \cos b - \sin a\, \sin b \\ 
\sin (a+b) &= \cos a\, \sin b + \sin a\, \cos b
\end{aligned}
\end{equation*}
are direct consequences of their definitions by exponential function and the addition theorem for the exponential function.

\item (***)  %%4
Find the values of $\sin i$, $\cos i$, $\tan (1 + i)$.

\item (***)  %%5
Use the addition formulas to separate $\cos (x + iy)$, $\sin (x + iy)$ in real and imaginary parts.

\item  %%6
For real $y$, show that every remainder in the series for $\cos y$ and $\sin y$ has the same sign as the leading term. 

\item  %%7
The smallest positive period of $e^{iz}$ is denoted by $2\pi$. 
Prove, for instance, that $3 < \pi < 2\sqrt{3}$. 

\item (***)  %%8
Find the real and imaginary parts of $\exp(e^z)$.

\item (***)  %%9
Determine all values of $2^i$, $i^i$, $(-1)^{2i}$.

\item  %%10
Show how to define the ``angles'' in a triangle, bearing in mind that they should lie between 0 and $\pi$. With this definition, prove that the sum 
of the angles is $\pi$.

\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


